Finite Group Databasebeta

← C2×C3×#63737 #13166854

Subgroups — C2×C3×#63737 #13166854

Order 1728.

1240 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×#157C211yesmaximal, characteristic
2C22×C3×#624C212yesmaximal
3C3×#63737C214yesmaximal, direct product factor
4C2×#63737C311yesmaximal, characteristic, direct product factor
5C23×C3×C3⋊D3C2211yescharacteristic
6C22×C3×#157—24no—
7C22×C3×#157C414yescomplemented
8C22×C3×#157C2212yes—
9C22×C32×Dic3—22no—
10C2×C3×#624—22no—
11C2×C3×#624—24no—
12C2×C3×#624C418yescomplemented
13C2×C3×#624C2214yes—
14C23×#157C2×C311yescharacteristic
15C22×C3×C4×D3—62no—
16C22×#624C2×C312yes—
17#63737C2×C314yesdirect product factor
18C23×C33D411yescharacteristic
19C22×C3×C3⋊D3C2×C412yes—
20C22×C3×C3⋊D3C2×C414yes—
21C22×C3×C3⋊D3D411yescharacteristic
22C22×C3×C3⋊D3D414yes—
23C22×C3×C3⋊D3Q812yes—
24C22×C3×C3⋊D3C2311yescharacteristic
25C2×C3×#157—28no—
26C2×C3×#157—24no—
27C2×C3×#157C2×C414yescomplemented
28C2×C32×Dic3—28no—
29C2×C32×Dic3—24no—
30C3×#624—28no—
31C3×#624—24no—
32C3×#624C2×C418yescomplemented
33C2×C3×#95—91nomaximal
34C23×C3⋊D3C22×C311yescharacteristic
35C23×C3×D3—62no—
36C22×#157—24no—
37C22×#157C3×C414yescomplemented
38C22×#157C22×C312yes—
39C22×C32×C4—62no—
40C22×C3×Dic3—22no—
41C22×C3×Dic3—22no—
42C22×C3×Dic3—42no—
43C2×C3×C4×D3—68no—
44C2×C3×C4×D3—64no—
45C2×C3×C4×D3—64no—
46C2×C3×C4×D3—68no—
47C2×#624—22no—
48C2×#624—24no—
49C2×#624C3×C418yescomplemented
50C2×#624C22×C314yes—
51C22×C33C22⋊C414yes—
52C22×C33C4⋊C412yes—
53C22×C33C2×D411yescharacteristic
54C2×C3×C3⋊D3C4214yes—
55C2×C3×C3⋊D3C22⋊C414yes—
56C2×C3×C3⋊D3C22⋊C414yes—
57C2×C3×C3⋊D3C4⋊C412yes—
58C2×C3×C3⋊D3C4⋊C418yes—
59C2×C3×C3⋊D3C22×C411yescharacteristic
60C2×C3×C3⋊D3C22×C412yes—
61C2×C3×C3⋊D3C2×D412yes—
62C2×C3×C3⋊D3C2×Q811yescharacteristic
63C3×#157—24no—
64C32×Dic3—28no—
65C23×C3×C4—92no—
66C23×C3×C4—91no—
67C22×C4×D3—62no—
68C3×#95—94no—
69C23×C32—22no—
70C23×C32—42no—
71C23×C32C3×D411yescharacteristic
72C22×C3⋊D3C2×C3×C412yes—
73C22×C3⋊D3C2×C3×C414yes—
74C22×C3⋊D3C3×D411yescharacteristic
75C22×C3⋊D3C3×D414yes—
76C22×C3⋊D3C3×Q812yes—
77C22×C3⋊D3C23×C311yescharacteristic
78C22×C3×D3—64no—
79C22×C3×D3—64no—
80C22×C3×D3—64no—
81C22×C3×D3—64no—
82C22×C3×D3—62no—
83C22×C3×D3—64no—
84C22×C3×D3—64no—
85C22×C3×D3—62no—
86C2×#157—28no—
87C2×#157—24no—
88C2×#157C2×C3×C414yescomplemented
89C2×C32×C4—64no—
90C2×C32×C4—68no—
91C2×C3×Dic3—24no—
92C2×C3×Dic3—28no—
93C2×C3×Dic3—28no—
94C2×C3×Dic3—24no—
95C2×C3×Dic3—48no—
96C2×C3×Dic3—44no—
97C3×C4×D3—68no—
98C3×C4×D3—616no—
99C3×C4×D3—68no—
100#624—28no—
101#624—24no—
102#624C2×C3×C418yescomplemented
103C2×#95—91noSylow
104C2×C33#9514yescomplemented
105C2×C33C2×C22⋊C412yes—
106C2×C33C2×C4⋊C411yescharacteristic
107C3×C3⋊D3#9514yescomplemented
108C3×C3⋊D3C2×C4211yescharacteristic
109C3×C3⋊D3C2×C22⋊C411yescharacteristic
110C3×C3⋊D3C2×C4⋊C412yes—
111C24×C3—91no—
112C23×D3—62no—
113C22×C3×C4—62no—
114C22×C3×C4—62no—
115C22×C3×C4—94no—
116C22×C3×C4—98no—
117C22×C3×C4—94no—
118C22×C3×C4—92no—
119C22×C3×C4—122no—
120C22×C3×C4—182no—
121C22×C3×C4—184no—
122C22×C3×C4—184no—
123C22×Dic3—22no—
124C2×C4×D3—68no—
125C2×C4×D3—64no—
126C2×C4×D3—64no—
127C2×C4×D3—68no—
128C22×C32—22no—
129C22×C32—24no—
130C22×C32—24no—
131C22×C32—24no—
132C22×C32—44no—
133C22×C32—44no—
134C22×C32—44no—
135C22×C32—42no—
136C22×C32C3×C22⋊C414yes—
137C22×C32C3×C4⋊C412yes—
138C22×C32C2×C3×D411yescharacteristic
139C2×C3⋊D3C3×C4214yes—
140C2×C3⋊D3C3×C22⋊C414yes—
141C2×C3⋊D3C3×C22⋊C414yes—
142C2×C3⋊D3C3×C4⋊C412yes—
143C2×C3⋊D3C3×C4⋊C418yes—
144C2×C3⋊D3C22×C3×C411yescharacteristic
145C2×C3⋊D3C22×C3×C412yes—
146C2×C3⋊D3C2×C3×D412yes—
147C2×C3⋊D3C2×C3×Q811yescharacteristic
148C2×C3×D3—68no—
149C2×C3×D3—64no—
150C2×C3×D3—64no—
151C2×C3×D3—68no—
152C2×C3×D3—64no—
153C2×C3×D3—68no—
154C2×C3×D3—68no—
155C2×C3×D3—62no—
156C2×C3×D3—62no—
157C2×C3×D3—62no—
158C2×C3×D3—62no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161#157—24no—
162C32×C4—68no—
163C3×Dic3—28no—
164C3×Dic3—28no—
165C3×Dic3—48no—
166C23×C4—92no—
167C23×C4—91no—
168#95—94no—
169C33C2×#9511yescharacteristic, complemented, Sylow
170C23×C3—22no—
171C23×C3—42no—
172C23×C3—91no—
173C23×C3—91no—
174C23×C3—94no—
175C23×C3—94no—
176C23×C3—92no—
177C23×C3—92no—
178C23×C3GO+(4,𝔽2)11yescharacteristic
179C22×D3—64no—
180C22×D3—64no—
181C22×D3—64no—
182C22×D3—62no—
183C22×D3—64no—
184C22×D3—64no—
185C22×D3—64no—
186C22×D3—62no—
187C2×C3×C4—64no—
188C2×C3×C4—68no—
189C2×C3×C4—64no—
190C2×C3×C4—68no—
191C2×C3×C4—98no—
192C2×C3×C4—94no—
193C2×C3×C4—124no—
194C2×C3×C4—128no—
195C2×C3×C4—184no—
196C2×C3×C4—188no—
197C2×C3×C4—184no—
198C2×C3×C4—188no—
199C2×Dic3—24no—
200C2×Dic3—28no—
201C4×D3—68no—
202C4×D3—616no—
203C4×D3—68no—
204C2×C32—22no—
205C2×C32—22no—
206C2×C32—22no—
207C2×C32—28no—
208C2×C32—48no—
209C2×C32—42no—
210C2×C32—42no—
211C2×C32—42no—
212C2×C32C3×#9514yescomplemented
213C2×C32C2×C3×C22⋊C412yes—
214C2×C32C2×C3×C4⋊C411yescharacteristic
215C3⋊D3C3×#9514yescomplemented
216C3⋊D3C2×C3×C4211yescharacteristic
217C3⋊D3C2×C3×C22⋊C411yescharacteristic
218C3⋊D3C2×C3×C4⋊C412yes—
219C3×D3—68no—
220C3×D3—62no—
221C3×D3—62no—
222C3×D3—62no—
223C3×D3—62no—
224C24—91no—
225C22×C4—62no—
226C22×C4—94no—
227C22×C4—98no—
228C22×C4—94no—
229C22×C4—92no—
230C22×C4—182no—
231C22×C4—184no—
232C22×C4—184no—
233C22×C3—22no—
234C22×C3—24no—
235C22×C3—24no—
236C22×C3—24no—
237C22×C3—42no—
238C22×C3—44no—
239C22×C3—44no—
240C22×C3—44no—
241C22×C3—92no—
242C22×C3—92no—
243C22×C3—94no—
244C22×C3—91no—
245C22×C3—91no—
246C22×C3—92no—
247C22×C3—94no—
248C22×C3—94no—
249C22×C3—98no—
250C22×C3D32⋊C414yes—
251C22×C3#377312yes—
252C22×C3C2×GO+(4,𝔽2)11yescharacteristic
253C2×D3—64no—
254C2×D3—62no—
255C2×D3—62no—
256C2×D3—68no—
257C2×D3—68no—
258C2×D3—64no—
259C2×D3—64no—
260C2×D3—68no—
261C2×D3—68no—
262C2×D3—62no—
263C2×D3—62no—
264C2×D3—62no—
265C2×D3—62no—
266C3×C4—68no—
267C3×C4—68no—
268C3×C4—128no—
269C3×C4—184no—
270Dic3—28no—
271C32—22no—
272C32—42no—
273C32C2×C3×#9511yescharacteristic, complemented
274C23—91no—
275C23—91no—
276C23—94no—
277C23—94no—
278C23—92no—
279C23—92no—
280C23C3×GO+(4,𝔽2)11yescharacteristic
281C2×C4—64no—
282C2×C4—68no—
283C2×C4—98no—
284C2×C4—94no—
285C2×C4—184no—
286C2×C4—188no—
287C2×C4—188no—
288C2×C4—184no—
289C2×C3—22no—
290C2×C3—28no—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—42no—
294C2×C3—48no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3—91no—
298C2×C3—91no—
299C2×C3—92no—
300C2×C3—94no—
301C2×C3#6373714yesdirect product factor
302C2×C3C2×D32⋊C412yes—
303C2×C3C2×#377311yescharacteristic
304D3—68no—
305D3—62no—
306D3—62no—
307D3—62no—
308D3—62no—
309C22—91no—
310C22—91no—
311C22—92no—
312C22—94no—
313C22—94no—
314C22—98no—
315C22—92no—
316C22—92no—
317C22—94no—
318C22C3×D32⋊C414yes—
319C22C3×#377312yes—
320C22C2×C3×GO+(4,𝔽2)11yescharacteristic
321C4—68no—
322C4—184no—
323C3—22no—
324C3—42no—
325C3C2×#6373711yescharacteristic, direct product factor
326C2—91no—
327C2—91no—
328C2—94no—
329C2—92no—
330C2C3×#6373714yesdirect product factor
331C2C2×C3×D32⋊C412yes—
332C2C2×C3×#377311yescharacteristic
subgroupG/H
center, hypercenterC23×C3GO+(4,𝔽2)
commutatorC3⋊D3C2×C3×C42
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, socleC23×C33D4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3⋊D3C23×C3
hypocenterC32C2×C3×#95
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C23×C3

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C3×#63737C2direct (N × H)
4. C2×#63737C3direct (N × H)
7. C22×C3×#157C4semidirect
12. C2×C3×#624C4semidirect
17. #63737C2×C3direct (N × H)
27. C2×C3×#157C2×C4semidirect
32. C3×#624C2×C4semidirect
37. C22×#157C3×C4semidirect
49. C2×#624C3×C4semidirect
88. C2×#157C2×C3×C4semidirect
102. #624C2×C3×C4semidirect
104. C2×C33#95semidirect
107. C3×C3⋊D3#95semidirect
169. C33C2×#95semidirect
212. C2×C32C3×#95semidirect
215. C3⋊D3C3×#95semidirect
273. C32C2×C3×#95semidirect
301. C2×C3#63737direct (N × H)
325. C3C2×#63737direct (N × H)
330. C2C3×#63737direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C3×#1572
C22×C3×#6242
C3×#637372
C2×#637373
C2×C3×#959